A Formal Logical Framework for Cadiag-2

648 Medical Informatics in a United and Healthy Europe K-P Adlassnig et al (Eds) IOS Press, 29 29 European Federation for Medical Informatics All rights reserved doi:12/978-1-675-44-5-648 A Formal Logical Framework for Cadiag-2 Agata CIABATTONI a, Thomas VETTERLEIN b,1, Klaus-Peter ADLASSNIG b a Institute for Discrete Mathematics and Geometry, Vienna University of Technology, Austria b Section on Medical Expert and Knowledge-Based Systems, Medical University of Vienna, Austria Abstract Cadiag-2 where Cadiag stands for computer-assisted diagnosis is an expert system based on fuzzy logic assisting in the differential diagnosis in internal medicine With its aid, it is possible to derive from possibly vague information about a patient s symptoms, signs, laboratory test results, and clinical findings conjectures about present diseases In this paper, we provide a mathematical formalization of the inferential mechanism of Cadiag-2 The aim is to have a formal logical calculus at hand which corresponds to the mode of operation of Cadiag-2 and which is among others needed to perform consistency checking of Cadiag-2 s medical knowledge base Keywords medical expert systems, Cadiag-2, fuzzy logic, logical calculus, internal medicine 1 Introduction Cadiag-1 and Cadiag-2 are computer-based medical consultation systems, developed at the University of Vienna Medical School (now Medical University of Vienna) since the 198 s (see, eg, [1]; for the performance, see, eg, [2]) Their aim is to support clinical differential diagnosis in the field of internal medicine Both systems are based on relationships between propositions about symptoms, signs, laboratory test results, and clinical findings (symptoms for short) on the one hand and diagnoses on the other hand In Cadiag-1, these propositions are treated as three-valued, that is, as being true, false, or undefined The relationships between these propositions can be formulated in the monadic fragment of first-order classical logic; the decidability of the latter makes it possible to check the consistency of Cadiag-1 s medical knowledge base; and actually 17 inconsistencies within 5, binary relationships have once been detected [] Precise information, however, is often not available to physicians to decide about a patient s disease In order to process vague information, the successor system Cadiag-2 was based on fuzzy logic [4] Note that Cadiag-2 relies entirely on fuzzy techniques An advantage of this choice, when compared to systems like DXplain [5], which are essentially based on probability theory, is the fact that Cadiag-2 inferences are always offered together with a justification which is easily comprehensible to the user 1 Corresponding Author: Spitalgasse 2, A-19 Vienna; E-Mail: thomasvetterlein@meduniwienacat

A Ciabattoni et al / A Formal Logical Framework for Cadiag-2 649 However, Cadiag-2 has not been explicitly formulated in the framework of a formal logic As a consequence, the problem of how to check the consistency of its rules is not yet well understood In this paper, we provide first steps towards solving this issue; we introduce a basic fuzzy-logical framework for Cadiag-2 2 The Cadiag-2 Inferential Mechanism We shall shortly describe the inferential mechanism of Cadiag-2 For a comprehensive description of the system, see, eg, [6] The knowledge base of Cadiag-2 consists of ifthen rules representing definitional, causal, statistical, and heuristic one-to-one or many-to-one relationships between symptoms and diseases On the basis of this general knowledge and the particular information referring to a patient (recounted history, observed signs, measured test results), the inference engine can draw conclusions We note that symptoms and diseases are not analysed with respect to their meaning, but are rather treated as pure propositions; what matters are their mutual relationships Propositions processed by Cadiag-2 An example of a proposition referring to a symptom might be suffering from strong abdominal pain It is obvious that the alternatives true and false to evaluate this proposition are not exhaustive Accordingly, Cadiag-2 considers the statements about symptoms as being vague Namely, to each symptom, there is associated a degree of presence, expressed by any element of the real unit interval [,1] The second class of propositions in Cadiag-2 refers to diagnoses It is often not or not yet possible to confirm or to exclude a diagnosis with certainty, thus, to each diagnosis, there is associated a degree of certainty, which is again a value in [,1] Let now, 1, m be all symptoms and 1,, n all diagnoses contained in Cadiag-2 s knowledge base Each such symbol is called a basic entity By the use of connectives, we can form compound entities; we have to our disposal conjunction, disjunction, and negation ~ For example, 1 ~ expresses the presence of the symptom 1 and the absence of the symptom Assume now that 1 and are assigned the truth values t1 and t, respectively Then we may calculate a truth value for 1 ~ as well, namely, we take mint, 1 1 t In general, if we are given an assignment of certain basic entities, we may extend it to as many compound ones as possible: Definition 1 An evaluation is a function v from a subset of the set of entities to the real unit interval [,1] such that the following holds: (i) If v s and v t, then v mins, t and v maxs, t ; (ii) if v or v, then v ; (iii) if v tand v is undefined, or v is undefined and v t, then v t ; (iv) if v t, then v ~ 1 t The input of one run of Cadiag-2 is an evaluation, called the initial evaluation and used to describe the state of a particular patient Then, inference rules are successively applied so as to generate a sequence of evaluations w, w, 1 2 Compared w

65 A Ciabattoni et al / A Formal Logical Framework for Cadiag-2 to its predecessor, each evaluation in this sequence encodes an increased amount of information about the patient The process terminates after finitely many, say l, steps, w l and is called the final evaluation The rules For each k 1,, l, w k 1 w k is the result of an application of a rule to Each rule, say R, originates from the knowledge base of Cadiag-2 and contains the following information: (i) a possibly compound entity, (ii) a basic entity, and (iii) the type of the logical relationship between and, which is one of the following: (c d ), where d (,1] Then R expresses that, which encodes, eg, a combination of symptoms, gives a hint to, which in turn encodes, eg, a diagnosis The implication holds the stronger the larger d is; d is called the confirmability degree (me) Then R expresses that and are mutually exclusive (ao) Then R expresses that if holds, then necessarily also holds Let R be of type (c d ), relating the entities and Then R is applied to w k1 and the confirmability degree d are follows The truth value t assigned to combined to one truth value b min t, d If then is not yet in the domain of wk1, we put wk b If otherwise w k1 and b, we put w k maxw k 1, b If w k1 and b1 or if w k1 1 and b, then w For the remaining basic entities, w coincides with w, k and the compound entities are defined according to Definition 1 Consider the following example of a rule of type (c ): IF suspicion of liver metastases by liver palpation THEN pancreatic cancer with the confirmability degree If, say, there is a clear suspicion of liver metastases by palpation, we evaluate the assumption of this rule with 1 An application of the rule then associates to the conclusion, unless there is better information available, a certainty degree The cases (me) and (ao) work similarly The rules are applied systematically one by one, but the order is arbitrary The process is completed if, by use of any of the rules, the evaluation remains unchanged k1 as k CadL the Logical Counterpart of Cadiag-2 In this section, we introduce CadL ( Cadiag logic ), a calculus adequate to formalise Cadiag-2 According to the ideas of Cadiag-2, CadL uses a concept well-known in fuzzy logic (see, eg, [7]): pairs consisting of a proposition and a rational truth value Definition 2 The atomic propositions of CadL are symbols The lattice propositions of CadL, denoted by F L 1, 2, are the expressions built up from the

A Ciabattoni et al / A Formal Logical Framework for Cadiag-2 651 atomic propositions by means of the connectives,, and ~ Moreover, the implications of CadL, denoted by FI are the expressions, where, FL Finally, FL F I are the propositions of CadL A graded proposition is a pair, t, where F and t,1 An entity in Cadiag-2 together with its image under an evaluation, corresponds to a graded proposition in CadL Definition The evaluation rules of CadL are, s, t,,, s t,,, s, t, u, s t, u, F and s, t,1, u(,1] for any L The manipulation rules are, d, t for any d, t, d t ~,1,1, ~ for any, F such that is atomic L, u, u,1,,, t ~,~ t A theory of CadL is a finite set T of graded propositions A proof from T is a finite sequence of graded propositions each of which is either in T or the conclusion of a rule whose assumptions are among the preceding elements of the proof The last entry in a proof from T is called provable from T The evaluation rules serve to determine the truth values associated to compound propositions; and the three manipulation rules mirror the three types of rules of Cadiag-2 We will now establish the correspondence between Cadiag-2 and CadL Given a Cadiag-2 knowledge base, we identify each basic entity with a unique atomic proposition of CadL, and each compound entity with the respective lattice proposition ofcadl Let us fix some initial evaluation of a run of Cadiag-2 We associate with w the following theory Tw of CadL: (i), w if FA is in the domain of w ; (ii),d for each rule in the knowledge base of type (c d ), where d (,1] ; (iii) ~,1 for each rule in the knowledge base of type (me); (iv) ~ ~,1 for each rule in the knowledge base of type (ao) Proposition 1 (completeness) Let be an entity in the domain of the final evaluation wl of a run of Cadiag-2 Then,, wl is provable in CadL from T w The converse direction is more delicate as not all the proofs in CadL correspond to a run of Cadiag-2 The reason is that when a new value is computed at the k -th step of w

652 A Ciabattoni et al / A Formal Logical Framework for Cadiag-2 a run of Cadiag-2, the previously obtained value for the same entity may become obsolete We strengthen the notion of a proof in CadL Definition 4 Call a proof of CadL strict if the following holds Let the i -th proof entry be derived by a rule, and let the j -th entry be among its assumptions, being of the form,t for some FL and t (,1] Then neither of the entries j1,,i1 is of the form,u, where is a subformula of ; and neither of the entries prior toi is, t for some t t ort Proposition 2 (soundness) Let P be a strict proof of CadL from Tw, and let, t be contained in P, where FL Then there is a run of Cadiag-2 with l steps such that w l t for some l l Propositions 1 and 2 together imply that (initial pieces of) runs of Cadiag-2 and strict proofs of CadL are in an exact mutual correspondence 4 Discussion and Conclusion We have shown that the mode of operation of Cadiag-2 can be represented in the framework of a formal logical calculus, called CadL Any general question about the inference of Cadiag-2 translates to a question about this logic Moreover, we are able to characterize the CadL system within the family of t- norm-based fuzzy logics, which are studied intensively We have furthermore prepared the ground for tackling one of the most important problems about the Cadiag-2 knowledge base, its consistency Acknowledgement This research was partially supported by the Vienna Science and Technology Fund (WWTF) Grant MA7-16 References [1] Adlassnig, KP, Kolarz, G (1982) Cadiag-2: Computer-assisted medical diagnosis using fuzzy subsets In Gupta, MM, Sanchez, E (eds) Approximate Reasoning in Decision Analysis, North-Holland Publishing Company, Amsterdam, 219 247 [2] Leitich, H, Adlassnig, K-P, Kolarz, G (22) Evaluation of two different models of semiautomatic knowledge acquisition for the medical consultant system CADIAG-II/RHEUMA Artificial Intelligence in Medicine 25():215 225 [] Moser, W, Adlassnig, K-P (1992) Consistency checking of binary categorical relationships in a medical knowledge base Artificial Intelligence in Medicine 4(5):89 47 [4] Zadeh, LA (1965) Fuzzy sets Information and Control 8():8 5 [5] Barnett, GO, Cimino, JJ, Hupp, JA, Hoffer, EP (1987) DXplain, An Evolving Diagnostic Decision-Support System Journal of the American Medical Association 258(1):67 74 [6] Adlassnig, K-P, Kolarz, G, Scheithauer, W, Grabner, H (1986) Approach to a hospital-based application of a medical expert system Medical Informatics 11, 25 22 [7] Hájek, P (1998) Metamathematics of Fuzzy Logic Kluwer, Dordrecht